The Monty Hall problem

July 28, 2010

I seem to have started something with all this talk of probability, risk and such (see my previous Aardvark post), and a number of people have asked me for another counter-intuitive probability puzzle. Well perhaps the most baffling is the Monty Hall problem. This is based on a game show where a contestant is faced with three doors. Behind two of the doors are goats; behind the third is a car – the star prize. The contestant is asked to pick a door, but before it’s opened, the host, who knows what is behind each of the doors, opens one of the other doors to reveal a goat. The contestant is then asked if they want to change their mind about the door they want to open. What’s the best thing to do? Most people reason that since it’s now down to two doors, which looks like a 50% chance of winning the car (after-all, it must be behind one of the doors), there’s no point in changing. Wrong! Changing actually increases your chance of winning to 66.6%. Why? Have a good think about it and when you want to know the answer just search for the Monty Hall problem on Google… the whole thing has become strangely controversial!


An Aardvark ruined my life & other improbable things

July 26, 2010

Alright, not very likely unless I live in Africa or fall into an aardvark enclosure in a zoo. But it does get me onto the question of probability. Let’s start with an easy one. What are the chances of any two people, out of a group of, say, 23, sharing the same birthday? Give up? It’s 50%. That seems sort of counter-intuitive doesn’t it? What about a group of 47? Now apart from the fact that 23 and 47 are funny numbers, which probably means there’s some tricky maths behind this, which is a bit of a clue, this time the answer is 95%. Amazing! Now before getting on to why this is important, try another one…

Mass poisoning

You are the worst sort of despot and like all tyrants you enjoy a good party. So you order 1000 bottles of the finest champagne. Unfortunately you discover that one of the bottles is poisoned, poisoned with a slow-acting and lethal in the smallest dose, no visible signs sort of poison that kills people stone dead after 12 hours. Luckily you have an unlimited supply of prisoners to use as ‘food tasters’. And luckily for them you are not a complete crackpot, so you decide to use the minimum number to check the champagne. What is the smallest number you require to be sure of identifying the poisoned bottle?

Before enlisting the help of a super computer, do you think the answer is a smallish number; a medium sized one, or something that looks like mass murder? Hold that thought.

Probability and risk

What this is all about is that we find it difficult to estimate probability, and in the real world, risk, when we are presented with complex situations. Even people who are in the talking-about-risk business find it difficult. For example, doctors are trained to communicate risks and they do these by using examples. Here’s some, try putting them in order from the most probable to the least probable:

  • Dying of any cause in the next year
  • Needing treatment after an accident with a bed or pillow
  • Death by murder
  • Being struck by lightening.

Admittedly this is a rather morbid selection, but, and you probably didn’t realise it, they are already in the right order (1:100, 1:2000, 1:100000, 1:10000000). But it was difficult, wasn’t it? So what’s going on? It’s that your brain is out of its depth, you just don’t know, and that’s because you’ve got no frame of reference; plus you have an unconscious habit of messing about with risk.

Interestingly researchers have found that:

  • People exaggerate rare risks and downplay common risks.
  • People have problems estimating risks for other than normal situations.
  • Personal risks are perceived to be greater than anonymous risks.
  • People underestimate risks they choose to take and overestimate risks they can’t control.

Risk and business

All this is worth worrying about because probability and risk have a part to play in how people behave in business situations. And how they behave, in particular senior managers, has a critical impact on the health of an organisation. Think about the last point above and the recent banking crisis. In fact all the points probably have a bearing on the financial crunch-mess. The lesson to learn is not to guess when you don’t understand the underlying facts or variables, a really important realisation for anyone who tends to conduct business by the seat of their pants. And something we business psychologists look out for when assessing high flyers.

PS: The answer to the poisoning question is 10 food tasters, as long as they take very small sips. With 10 people there are 1024 unique combinations of people and bottles. More than enough to sort out the problem and, fantastic news, each person will have at least a 50% chance of living!

PPS: OK, a despot who was also a mathematical genius could get the figure down to 8 by avoiding the combinations where everyone sips from all the bottles; and those where all but one sip from all the bottles.

Photo credit: jscreationzs/FreeDigitalPhotos.net


How to beat the odds

April 11, 2010

Fascinating stuff probability, get your head round this: Generally speaking when you make a choice it can be a good one, a middling (average) one or a bad one. And it follows that if you have nothing else to go on your chance of making a below average choice is one in three. However if you assess an opportunity and reject it, and accept the next opportunity that is at least as good as the one you rejected, your chance of making a below average choice is now one in nine (a one-in-three followed by another one-in-three). And turning it the other way up, looking at above average choices, this approach must improve your chances of making a good call. In fact if the opportunity you rejected was above average then you are bound to make a good choice. And working through all the other combinations, without using the ‘assess it & reject it’ approach, your chances of making a below average or any other choice are stuck at one-in-three; whereas with the ‘rule’, as there is more than one way of making a good choice, your chances of making a good one are now six-in-nine. So using this approach your overall chance of making a good choice has doubled…

PS: This only works when you can learn something about the relative merits of different choices. Sadly it will not help you win the Lottery.


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